Three in One: Boom

Let $f(x)$ be a function continuous for all $x\in \mathbb R$ except at $x=0$ such that:

• $f'(x) < 0, \forall x \in(-\infty,0)$
• $f'(x) > 0, \forall x \in(0,\infty)$
• $\lim_{x \to 0^{+}} f(x) = 2$
• $\lim_{x \to 0^{-}} f(x) = 3$
• $f(0) = 4$

And that:

$\begin{cases} \displaystyle 2\lim_{x \to 0}f(x^3 - x^2) =\displaystyle \mu \lim_{x \to 0}f(2x^4 - x^5) \\ \displaystyle \lim_{x \to 0^{+}} \dfrac{f (-x) x^2} {\left \{\frac{1-\cos x}{\lfloor f(x) \rfloor} \right \}} = \lambda \\ \displaystyle \lim_{x \to 0^{-}} \left (\left \lfloor 3f \left(\frac{x^3 - \sin^{3}(x)}{x^4}\right) \right \rfloor - f \left(\left \lfloor \frac{\sin(x^3)}{x} \right \rfloor \right) \right) = \varphi \end{cases}$

Find $\mu \cdot \lambda \cdot \varphi$

Notations:

• $\lfloor \cdot \rfloor$ denotes the floor function .

• $\{ \cdot \}$ denotes the fractional part function .